Conduction Shape Factor Models for Three-Dimensional Enclosures
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Bibliographic record
Abstract
Analytical models are presented for conduction shape factors for three-dimensional regions formed between an isothermal, arbitrarily shaped body and its concentric, arbitrarily shaped surrounding enclosure. The model is based on the exact solution for the concentric spheres, and two methods are developed to predict the effective gap spacing. The models are validated using existing numerical data from the literature and data from simulations performed using a commercial computational fluid dynamics software package. The models are shown to be in excellent agreement with the data for all enclosures with geometrically similar boundary shape, within 3 % rms. For enclosures formed between different boundary shapes, the models are shown to be accurate within 5 % rms when the minimum aspect ratio, that is, the smallest outer boundary dimension over the largest inner dimension, is greater than 1.5. Nomenclature A = area, m2 a, b, c = cuboid side dimensions, m d = diameter, m k = thermal conductivity, W/mK L = general characteristic length, m m = combination parameter n = outward facing normal vector Q = total heat flow rate, W R = thermal resistance, K/W r = general radial coordinate S = conduction shape factor, m SL = dimensionless conduction shape factor, SL/Ai s = cube side length, m T = temperature, ◦C V = enclosed volume, m3 x, y, z = Cartesian coordinates δ = gap spacing, (do − di)/2, m ρ = local radial position, m φ, θ = spherical coordinates ψ = dimensionless temperature rise Subscripts e = effective i = inner o = outer ∞ = full-space limit
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it